Entry 1075 · 2026-08-20 · marici.Benincasa · research

1075 — The Full Bubble Cech Nerve Retains the Intrinsic Corner Line

Hard-to-vary claim

Entry 1073's rank-one corner cohomology is not an artifact of replacing the full three-divisor Čech nerve by a spanning tree. The complete labelled complex has

$$\boxed{ (\dim H^0,\dim H^1,\dim H^2)=(3,1,0). }$$

The spanning-tree presentation is a contraction of an acyclic pair–triple block and is therefore quasi-isomorphic to the full nerve.

Frozen full complex

Retain the three source spurious kernels

$$K_s=\ker M_s, \qquad s=6,7,8,$$

inside the common six-master fiber $V$. Every pairwise sum fills the ambient fiber in the places used below, so the labelled Čech terms are

$$C^0=K_6\oplus K_7\oplus K_8,$$

$$C^1=V_{67}\oplus V_{68}\oplus V_{78},$$

$$C^2=V_{678}.$$

With the ordered occurrence orientation $6<7<8$, define

$$d_0(v_6,v_7,v_8) = (v_7-v_6,\;v_8-v_6,\;v_8-v_7),$$

and

$$d_1(a_{67},a_{68},a_{78}) = a_{78}-a_{68}+a_{67}.$$

The exact checker verifies

$$d_1d_0=0.$$

No pair or triple term was introduced after inspecting the rank: these are the complete labelled intersections of the three frozen divisors.

Exact ranks

The cochain dimensions are

$$(\dim C^0,\dim C^1,\dim C^2)=(14,18,6).$$

Over both replication primes,

$$\operatorname{rank}d_0=11, \qquad \operatorname{rank}d_1=6.$$

Therefore

$$\dim H^0 =14-11=3,$$

$$\dim H^1 =(18-6)-11=1,$$

and

$$\dim H^2 =6-6=0.$$

Comparison with the spanning-tree presentation

Entry 1073 retained two pair comparisons,

$$(v_6-v_7,\;v_7-v_8),$$

and omitted the dependent third pair together with the triple term. The full calculation proves that this omission removes one surjective pair-to-triple block:

$$C^1_{\rm redundant}\longrightarrow C^2$$

of equal rank six. That block is acyclic. Contracting it leaves the two-term complex of Entry 1073 without changing $H^0$ or $H^1$.

Hence

$$\boxed{ H^1\simeq\mathbb Q }$$

is intrinsic to the labelled kernel diagram, not to the selected spanning tree.

Interpretation

Combined with Entry 1074, the full picture is now:

• $H^0$ is the rank-three common regularity space selected by the source;
• $H^1$ is a rank-one obstruction to independently gluing local spurious-divisor data;
• $H^2=0$, so there is no additional triple-overlap obstruction;
• the physical source boundary map lands diagonally in $H^0$ and does not activate $H^1$;
• all terms are supported on the existing three divisors and their frozen triple flat.

No new carrier datum is indicated.

Next falsifier

Move to the nearest independent polylogarithmic arrangement with a source-defined common boundary prescription. Construct its complete labelled Čech nerve before contracting it. Test whether the physical boundary again factors through $H^0$, or whether the source canonically activates higher corner cohomology.

Durable verification

• checker: research/benincasa/check_bubble_parabolic_complex.rs;
• packet: research/benincasa/bubble-full-cech-nerve.json;
• replication primes: $32003,32009$;
• allocator claim: seqclaim-ef651dbb7b3bfb5a44f27cef;
• epistemic event: ev-000000000755-83dfffc5-2af2-4ecf-98da-2cf07551152c.

View source on GitHub